Using Hypergraph to Design Perfect Secret Sharing Schemes for General Access Structures

• Main Authors: Yi-Chun, 王怡君
• Other Authors: Justie Su-Tzu Juan
• Format: Others
• Language: en_US
• Published: 2007
• Online Access: http://ndltd.ncl.edu.tw/handle/82627995009795757195

碩士 === 國立暨南國際大學 === 資訊工程學系 === 95 === A secret sharing scheme is a method to distribute a secret, also called master key, among a set of participants, such that only qualified subsets of the participants can recover the secret. A secret sharing scheme is perfect if any unqualified subset obtains no information regarding the master key. The collection of qualified subsets is called the access structure. In a hypergraph, if the size of any edge is equal to $r$, the hypergraph is called an $r$-uniform hypergraph. Similarity, if the size of any edge is equal to either $r_1$ or $r_2$ in a hypergraph, the hypergraph is called an ($r_1, r_2$)-uniform hypergraph. And, if the possible size of any edge is $r_1, r_2$ or $r_3$ in a hypergraph, the hypergraph is called an ($r_1, r_2, r_3$)-uniform hypergraph. Given any hypergraph $G$, a $G$-based access structure is an access structure which using $G$ present the access structure, where a vertex denote a participant and the edge set denote the minimal access structure of a secret sharing scheme. In this thesis, we propose four perfect secret sharing schemes for $r$-uniform, ($r_1, r_2$)-uniform, ($r_1, r_2, r_3$)-uniform and general hypergraph-based access structures (called $r$-HA, ($r_1, r_2$)-HA, ($r_1, r_2, r_3$)-HA and G-HA scheme respectively). Moreover, we modify G-HA scheme such that it: 1. can verify the shares (verification) and detect the cheater (detection), 2. can be reused, that is, will be multi-use secret sharing scheme, and 3. will be a multi-use secret sharing scheme with verification and detection. At last, this thesis shows that $r$-HA, ($r_1, r_2$)-HA, ($r_1, r_2, r_3$)-HA and G-HA schemes are all more efficient secret sharing scheme than the scheme be hold by Tochikubo, Uyematsu and Matsumoto in 2005 for the respective same access structure.